Tatel, Circuit Judge:

More than a decade ago, the parents of six children, ages three to six, sued the District of Columbia, alleging that it was violating the "Child Find" requirement of the Individuals with Disabilities Education Act by failing to provide special education to their children and hundreds of other preschoolers with disabilities. The district court certified the suit as a class action under Federal Rule of Civil Procedure 23, found the District liable, and entered a comprehensive injunction designed to bring the District into compliance with IDEA. On appeal, the District argues that the case has become moot because the six named plaintiffs are no longer toddlers with a stake in the requested relief. The District also challenges the class certification and argues that the injunction exceeds the district court's authority. For the reasons set forth in this opinion, we affirm in all respects.

I.

For much of this nation's history, children with disabilities "were either totally excluded from schools or were sitting idly in regular classrooms awaiting the time when they were old enough to drop out." Board of Education of Hendrick Hudson Central School District v. Rowley, 458 U.S. 176, 179, 102 S.Ct. 3034, 73 L.Ed.2d 690 (1982) (alteration and internal quotation marks omitted). Faced with this "pervasive and tragic academic stagnation," Congress passed the Education of the Handicapped Act of 1975 (EHA). Endrew F. ex rel. Joseph F. v. Douglas County School District RE-1, ___ U.S. ___, 137 S.Ct. 988, 999, 197 L.Ed.2d 335 (2017). That "ambitious" law, Rowley, 458 U.S. at 179, 102 S.Ct. 3034, which applied to public schools receiving federal funds, sought to provide all children with disabilities a "free appropriate public education ... tailored to [their] unique needs," id. at 180, 102 S.Ct. 3034.

Fifteen years later, finding that EHA implementation had "been impeded by low expectations" and resource constraints, 20 U.S.C. § 1400(c)(4)-(7), Congress strengthened the statute and renamed it the Individuals with Disabilities Education Act (IDEA), see Pub. L. No. 101-476, § 901, 104 Stat. 1103 (1990). This time Congress drew on its authority under the Spending Clause to offer states a deal: in exchange for additional federal funding, they would "pledge[] to comply" with a series of requirements designed to ensure that each student receives a "free appropriate public education," or FAPE. Endrew F., 137 S.Ct. at 993; see U.S. CONST., Art. I, § 8, cl. 1. Among the most important of these requirements, the "Child Find" provision obliges states to develop a "practical method" to track which children are receiving special education services and to ensure that all children "who are in need of special education and related services ... are identified, located, and evaluated" within a timeframe set by the state — 120 days in this case. 20 U.S.C. § 1412(a)(3)(A); see 20 U.S.C. § 1414(a)(1)(C)(i)(I) (authorizing states to "establish[] a timeframe within which the evaluation must be conducted"); D.C. Code § 38-2561.02(a)(1) (requiring an evaluation "within 120 days from the date the student was referred for an evaluation"). Another requirement, the "smooth and effective transition" condition, obliges states to provide a seamless transition when three-year-olds move from "early intervention" programs (governed by IDEA Part C) to preschool (governed by IDEA Part B). 20 U.S.C. §§ 1412(a)(9), 1435(a)(8)(A), 1437(a)(9); 34 C.F.R. § 303.209. The transition between these programs qualifies as "smooth and effective" if, among other things, it begins at least ninety days before the child's third birthday, delivers uninterrupted services, and involves both Part B and C personnel. 20 U.S.C. § 1412(a)(9); 34 C.F.R. § 303.209. In the District of Columbia, which IDEA defines as a state, see 20 U.S.C. § 1401(31), and which receives millions of dollars of IDEA funding each year, early intervention programs are run by the Office of the State Superintendent of Education and preschool programs by the District of Columbia Public Schools (DCPS).

In 2005, the parents of six children, ages three to six, sued the District, alleging a "pervasive and systemic" breakdown in the school system's Child Find program. D.L. v. District of Columbia, No. 05-cv-1437, ECF No. 1, at 3 (D.D.C. July 21, 2005). According to the complaint, the District was failing to identify large numbers of disabled children and delivering inadequate and delayed services to many others. These deficiencies, the parents argued, were depriving "hundreds" of preschoolers of their right to a FAPE. Id.

The district court, Judge Royce C. Lamberth, certified the suit as a class action in 2006. D.L. v. District of Columbia, 237 F.R.D. 319 (D.D.C. 2006). The class definition was broad: "All children [between three and five] who are or may be eligible for special education and related services, who live in, or are wards of, the District of Columbia" and whom the District had failed or would fail to "identify, locate, evaluate or offer special education and related services." Id. at 324-25. Four years later, the parties each moved for summary judgment. After reviewing the record, the district court granted summary judgment to the parents with respect to their claims up to and through 2007 and scheduled a bench trial on all remaining claims. D.L. v. District of Columbia, 730 F.Supp.2d 84, 95, 98 (D.D.C. 2010). During that trial, the court heard two days of testimony from statisticians, school district staff, and experts in education policy and early childhood development.

After trial but before the district court issued its decision, the Supreme Court decided Wal-Mart Stores, Inc. v. Dukes, which held that "one of the most expansive class[es] ever" was too broad to meet the requirements of Federal Rule of Civil Procedure 23. 564 U.S. 338, 342, 131 S.Ct. 2541, 180 L.Ed.2d 374 (2011). Relying on Wal-Mart, the District moved to decertify the class, arguing that it was similarly overbroad. Although the parents insisted that the class complied with Wal-Mart, they suggested that if the court had any doubt it should recertify the class as four subclasses of children whom the District had failed to (1) identify, (2) timely evaluate, (3) offer a timely determination of eligibility for special education and related services, and (4) provide a smooth transition from early intervention programs to preschool. D.L., ECF No. 271-2, at 7-8 (Aug. 18, 2011). Satisfied that the certified class complied with Wal-Mart, however, the district court deemed subclasses unnecessary. D.L. v. District of Columbia, 277 F.R.D. 38, 46-47 (D.D.C. 2011).

The court then found the District liable for violating its Child Find obligations and failing to ensure a "smooth and effective transition" for toddlers entering preschool. D.L. v. District of Columbia, 845 F.Supp.2d 1, 21-23 (D.D.C. 2011). These violations, Judge Lamberth observed, deprived "some of our most vulnerable citizens" of services in the "first few years" of their lives, a "narrow window of opportunity in which special education, tailored to the child's particular needs, can work a miracle." Id. at 5. Based on these findings, the court entered a comprehensive injunction that set compliance benchmarks and required annual improvement in the numbers of children identified as needing, evaluated for, and offered special education and related services.

The District appealed, and this court vacated the class certification order. D.L. v. District of Columbia, 713 F.3d 120, 121 (D.C. Cir. 2013). Citing Wal-Mart, we held that a class defined by reference "to the District's pattern and practice of failing to provide FAPEs speaks too broadly because it constitutes only an allegation that the class members `have all suffered a violation of the same provision of law.'" Id. (quoting Wal-Mart, 564 U.S. at 350, 131 S.Ct. 2541). We noted that the parents "appeared to recognize [this] problem," and had proposed subclasses tied to failures in four distinct administrative functions. Id. at 128. Rather than deciding whether those subclasses satisfied Wal-Mart, we remanded to the district court to consider that question in the first instance.

The district court then certified the same four subclasses the parents had proposed: (1) disabled three-to-five-year-olds whom the District failed to identify for the purpose of offering special education services; (2) disabled three-to-five-year-olds whom the District failed to give an initial evaluation within 120 days of being referred for special education services; (3) disabled three-to-five-year-olds whom the District failed to give an "eligibility determination" — i.e., a decision as to whether they qualify for IDEA services — within 120 days of being referred; and (4) all children who transitioned from early intervention to preschool programs, and whom the District denied a "smooth transition" by age three. This court denied the District's petition for interlocutory review, In re District of Columbia, No. 13-8009, Doc. No. 1477562 (D.C. Cir. Jan. 30, 2014), and the case once again advanced to summary judgment, where the district court entered judgment for the District on all claims concerning subclass two, and then on to a bench trial. D.L. v. District of Columbia, 109 F.Supp.3d 12, 36 (D.D.C. 2015). After considering testimony from seventeen witnesses and reviewing hundreds of exhibits, the district court issued a 130-page opinion finding the District liable for violating IDEA. D.L. v. District of Columbia, 194 F.Supp.3d 30 (D.D.C. 2016).

The district court's findings were stark. It found that the District was failing to identify between 98 and 515 children a month — some two percent of preschoolers with disabilities who should have been located and offered special education services. Id. at 48. This placed the District's Child Find performance below jurisdictions with comparable rates of childhood disability, such as Arkansas, Kentucky, and Puerto Rico. Id. at 48, 53. In addition, the court found that the District was failing to provide a "smooth and effective transition" to almost 30 percent of disabled toddlers, id. at 63, and despite having the "longest period of time in the country" to decide whether children qualify for special education services, was missing the deadline for issuing eligibility determinations approximately 20 percent of the time, id. at 58-59.

Acknowledging that the District had improved its Child Find program since 2007, when it had "the lowest percentage" of special education enrollment in the United States, the court stressed that "the District ha[d] yet to attain a period of sustained compliance." Id. at 78, 98. Indeed, the court observed, the numbers of children receiving special education had fallen by 15 percent in 2013 and 2014 when the District lacked "an enrollment benchmark" — i.e., a target number of children who should be enrolled in special education and related services. Id. at 51. Given these deficiencies, the court concluded that injunctive relief was necessary and, drawing on its "broad authority to grant `appropriate' relief,'" Forest Grove School District v. T.A., 557 U.S. 230, 239, 129 S.Ct. 2484, 174 L.Ed.2d 168 (2009) (construing 20 U.S.C. § 1415(i)(2)(C)(iii)), crafted remedies for the three remaining subclasses.

For subclass one — children the District was failing to identify — the court set an 8.5 percent enrollment target, a figure drawn from national rates of special education enrollment and expert testimony concerning risk factors unique to Washington, D.C. To reach that target, the court required the District to increase enrollment rates by half a percent each year. For subclass three — children denied timely eligibility determinations — the court ordered the District to meet the statutory deadline 95 percent of the time and to improve its performance annually until it reached that level. Similarly, for subclass four — toddlers transitioning to preschool — the court required annual improvement, with an ultimate goal of 95 percent compliance. The court also imposed a range of "programmatic" remedies designed to improve the District's methods of finding and tracking children in its system. D.L. v. District of Columbia, 194 F.Supp.3d 30, 101-03. These remedies included requirements that the District establish databases, disseminate information to parents, and report its progress to the court.

On appeal, the District challenges none of the district court's basic findings: that it was failing to identify children with disabilities, that it often missed the deadline for issuing eligibility determinations, and that it was providing a rocky transition to toddlers entering preschool. Instead, it argues that: (1) the case is moot because by the time the district court certified the subclasses in 2013 each named plaintiff was over age five, (2) class certification was improper under Wal-Mart, and (3) the injunction was unauthorized by IDEA and unsupported by the evidentiary record.

Before considering these arguments, we think it helpful to note that the parents who brought this case are not the only ones concerned with the District's IDEA compliance. Since at least 1997, the U.S. Department of Education, which oversees state performance under IDEA, has repeatedly warned the District that it was neglecting its Child Find obligations. See id. at 72-78 (documenting the Department's correspondence with the District since the mid-1990s); see also 20 U.S.C. § 1416(d)-(e) (authorizing the Secretary of Education to review state IDEA compliance and to withhold federal funds). In 1998, the Department "entered into a Compliance Agreement with DCPS mandating full compliance with the requirements of Part B of the IDEA." D.L., 194 F.Supp.3d at 76. In the two decades since, the Department has nonetheless regularly listed DCPS as a school district that "needs intervention." Id. at 73-79. According to the Department, the District has been especially deficient in its duty to timely evaluate children referred for special education by a parent, teacher, or pediatrician. Id. at 73. The Department even withheld a portion of the District's funding in 2009. Id. at 77. Although the record here reveals no specific link between the Department's actions and this case, the subclass two complaints, which focus on the same problem with timely evaluation, have been resolved. See D.L. v. District of Columbia, 109 F.Supp.3d 12, 36 (D.D.C. 2015) (granting summary judgment as to subclass two). This case now involves identification (subclass one), eligibility determinations (subclass three), and transition to preschool (subclass four).

II.

Beginning with mootness, we start from a point on which the parties agree: when the district court certified subclasses, the named plaintiffs' individual claims for injunctive relief were moot because, by that time, each child was older than five and, according to the District, had received special education services. The District argues that this rendered the dispute non-justiciable. According to the parents, however, two exceptions to the mootness doctrine apply: first, a "relation back" exception, which permits class actions to proceed when a named plaintiff's individual claim becomes moot only after a district court's error; and second, the "inherently transitory" exception, which applies to claims so fleeting that "the trial court will not have even enough time to rule" on class certification before the named plaintiff's claim expires. Genesis Healthcare Corp. v. Symczyk, ___ U.S. ___, 133 S.Ct. 1523, 1530-31, 185 L.Ed.2d 636 (2013) (citation and internal quotation marks omitted).

The Supreme Court articulated the first of these exceptions in United States Parole Commission v. Geraghty, 445 U.S. 388, 100 S.Ct. 1202, 63 L.Ed.2d 479 (1980). In that case, a prisoner challenged the constitutionality of the federal parole guidelines, the district court erroneously denied the prisoner's request for class certification, and the prisoner was released before the Third Circuit could correct the error. Id. at 394, 100 S.Ct. 1202. The Court held that where an action "would have acquired the independent legal status" of a class action "but for the district court's erroneous denial of class certification, a corrected ruling on appeal `relates back' to the time of the erroneous denial." Genesis Healthcare, 133 S.Ct. at 1530 (describing the rule in Geraghty).

The parents argue that this case is just like Geraghty. We agree. Here, the district court ruled that an overly broad class satisfied Wal-Mart, an error this court corrected on appeal. See supra 718-19. Like the plaintiffs in Geraghty, the parents had live claims when they sought certification, and but for the district court's error, could have obtained proper class certification before their individual claims became moot. Under Geraghty, then, the case remains justiciable.

Resisting this conclusion, the District argues that Geraghty applies only when a court erroneously denies class certification. In support, the District points out that the Supreme Court stated in Geraghty that its holding was "limited to the appeal of the denial of the class certification motion." Geraghty, 445 U.S. at 404, 100 S.Ct. 1202. The District reads too much into the word "denial." The point in Geraghty was that claims relate back when a trial court's error prevents a class from gaining independent status under Rule 23. Whether that error is the erroneous denial of class certification (as in Geraghty) or the erroneous certification of an excessively broad class (as here) makes no difference. What matters is that the named plaintiffs' claims became moot — and their class therefore never "acquired ... independent legal status," Genesis Healthcare, 133 S.Ct. at 1530 — due to the district court's mistake. In other words, but for the district court's error — certifying an overly broad class — the parents' claims would not have become moot. There is no legally relevant difference between this case and Geraghty.

The District insists that the parents could have avoided this entire problem by finding new toddler-plaintiffs in 2013 when this court remanded the case to the district court. This misses the point: when the relation back doctrine applies, as it does here, named plaintiffs have no obligation to find new class representatives even if they could.

Permitting relation back in this case is also consistent with Rule 23's purpose. As the Supreme Court explained in Geraghty, the "justifications that led to the development of the class action include ... the provision of a convenient and economical means for disposing of similar lawsuits[] and the facilitation of the spreading of litigation costs among numerous litigants with similar claims." 445 U.S. at 402-03, 100 S.Ct. 1202 (citing Advisory Committee Notes on Fed. R. Civ. P. 23). Those interests are served by a rule allowing class claims to proceed when a district court erroneously certifies too broad a class while plaintiffs' claims are live, only to be reversed and instructed to consider smaller subclasses containing exactly the same children. By contrast, Rule 23's purpose would be disserved by a rule, advocated by the District, requiring parents to find new named plaintiffs at every turn of inevitably protracted class litigation.

Ultimately, the District's argument runs counter to the Supreme Court's instruction in Geraghty that "Art[icle] III mootness doctrine" has a "flexible character." Id. at 400, 100 S.Ct. 1202. Mootness is a pragmatic doctrine meant to limit "judicial power to disputes capable of judicial resolution." Id. at 396, 100 S.Ct. 1202; cf. Campbell-Ewald Co. v. Gomez, ___ U.S. ___, 136 S.Ct. 663, 669, 193 L.Ed.2d 571 (2016) ("A case becomes moot ... `only when it is impossible for a court to grant any effectual relief whatever to the prevailing party.'" (quoting Knox v. Service Employees, 567 U.S. 298, 132 S.Ct. 2277, 2287, 183 L.Ed.2d 281 (2012))). In this case, the mootness issue stems neither from the lack of real dispute nor from any deficiency in the parents' advocacy, but rather from judicial error. The separation of powers concerns that animate justiciability jurisprudence are absent in this context. In Geraghty, the Court emphasized that the two elements of a justiciable controversy — "sharply presented issues in a concrete factual setting and self-interested parties vigorously advocating opposing positions" — can "exist with respect to ... class certification ... notwithstanding the fact that the named plaintiff's claim on the merits has expired." 445 U.S. at 403, 100 S.Ct. 1202. Both elements remain present here.

Citing Genesis Healthcare, 133 S.Ct. at 1523, the District argues that "[n]o exception to the mootness doctrine allowed" certification in 2013. Appellants' Br. 30. The question in Genesis Healthcare was whether the relation back doctrine applied when an individual claim under the Fair Labor Standards Act (FLSA) expired before certification of a "collective action," an opt-in procedure established by FLSA for litigating multiple claims. See 29 U.S.C. § 216(b). The Court held that it did not and that the action was therefore non-justiciable. Genesis Healthcare, 133 S.Ct. at 1530-32. In reaching this conclusion, however, the Court relied not just on relation back cases, but "[m]ore fundamentally" on the distinction between FLSA collective actions, which do not "produce a class with an independent legal status," and Rule 23 class actions, which do. Id. at 1530. The outcome in Genesis Healthcare thus hinged on the unique features of the FLSA cause of action. In this case, by contrast, we are clearly in Rule 23-land and guided by Geraghty.

Having concluded that the relation back doctrine applies, we have no need to consider whether the parents' claims also fall under the "inherently transitory" exception to mootness. Id. at 1530-31. We thus turn to the District's challenge to class certification.

III.

Federal Rule of Civil Procedure 23 requires plaintiffs to show that:

If the action satisfies these prerequisites, plaintiffs must then demonstrate that their proposed class falls into one of the categories of class actions listed in Rule 23(b). In this case, the district court certified subclasses under Rule 23(b)(2), which applies when the defendant "has acted or refused to act on grounds that apply generally to the class, so that final injunctive relief or corresponding declaratory relief is appropriate respecting the class as a whole." Courts may certify classes under this provision "only when a single injunction or declaratory judgment would provide relief to each member of the class." Wal-Mart, 564 U.S. at 360, 131 S.Ct. 2541.

The Supreme Court interpreted these requirements in Wal-Mart, in which named plaintiffs seeking to represent 1.5 million women sued the retail giant Wal-Mart under Title VII of the Civil Rights Act of 1964, alleging endemic sex discrimination in pay and promotions across the company's "approximately 3,400 stores." Id. at 342, 131 S.Ct. 2541. The district court certified a class of "all women employed at any Wal-Mart domestic retail store at any time [in the prior thirteen years] who [had] been or may be subjected to" the company's challenged policies and practices. Id. at 346, 131 S.Ct. 2541 (alterations and internal quotation marks omitted).

Explaining that the class lacked commonality, the Court reversed. "Commonality requires the plaintiff to demonstrate that the class members have suffered the same injury," the Court observed. Id. at 349-50, 131 S.Ct. 2541 (citation and internal quotation marks omitted). Yet there are multiple theories of Title VII liability — that statute can, "for example, ... be violated ... by intentional discrimination, or by hiring and promotion criteria that result in disparate impact, and by the use of these practices on the part of many different superiors in a single company." Id. at 350, 131 S.Ct. 2541. Given this, the Court concluded, the assertion that Wal-Mart had violated Title VII in one way or another as to each employee did not demonstrate "that all their claims [could] productively be litigated at once." Id. Instead, plaintiffs needed a "common contention... of such a nature that it is capable of classwide resolution — which means that determination of its truth or falsity will resolve an issue that is central to the validity of each one of the claims in one stroke." Id.

As mentioned above, our court has already considered Wal-Mart's applicability to this case. We rejected the first certified class, which included all three-to-five-year-olds allegedly denied a FAPE, because it spanned "different policies and practices at different stages of the District's Child Find and FAPE process." D.L., 713 F.3d at 127. "For some plaintiffs," we explained, "the alleged harm suffered is due to the failure of the District to have an effective intake and referral process; for others the alleged harm is caused by the District's failure to offer adequate and timely education placements ...; for still others, the cause is the absence of a smooth and effective transition...." Id. at 128. Seeming to agree that narrower subclasses could resolve the commonality problem, the District argued that the class violated Wal-Mart because it "cover[ed] failures in four distinct administrative functions: (1) identification of a child ..., (2) location of that child, (3) evaluation for potential services, and (4) if necessary, provision of services." Id. (quoting Appellant's Br. 29).

On remand, the district court addressed just this defect, certifying subclasses tied to separate phases of the Child Find process. Three of those subclasses consist of three-to-five-year-olds whom the District had failed to (1) identify, (2) evaluate within 120 days of referral, and (3) provide an eligibility determination within 120 days of referral. The fourth subclass contains all children with disabilities denied smooth transitions from early intervention to preschool programs. Although these four subclasses appear responsive to both our 2013 opinion and the District's concerns, this time around the District argues that they too are insufficient under Wal-Mart. Once again, then, we review the district court's certification order for abuse of discretion. Hartman v. Duffey, 19 F.3d 1459, 1471 (D.C. Cir. 1994) ("[W]e review a class certification decision conservatively only to ensure against abuse of discretion or erroneous application of legal criteria."). We find none.

The three subclasses at issue here (recall that the court resolved subclass two claims before trial) are each defined by reference to a "uniform policy or practice" governing a specific stage of the special education process. D.L., 713 F.3d at 127. Whereas before the parents' claims were united only by a shared allegation of IDEA liability, now the suit has subclasses cast around "common harm[s]," id. at 128, susceptible to common proof, and curable by a "single injunction," Wal-Mart, 564 U.S. at 360, 131 S.Ct. 2541.

Take subclass one, children with disabilities whom the District failed to find. These children identified a common harm, namely, denial of a FAPE due to a deficient and poorly implemented Child Find policy. This contention, as is evident from the district court's findings, is subject to common proof: after reviewing the evidence, the court found that the District was failing to identify 98 to 515 children a month. This violation of the statute can, as is also evident from the district court's decision, be remedied by a single order, i.e., an injunction requiring the District to identify 0.5 percent more children each year until it reaches 8.5 percent enrollment.

As with subclass-one parents, the parents of subclass-three children allege a common harm: contrary to the District's own policy, their children had not received eligibility determinations within 120 days of being referred for a disability evaluation. Again, this contention can be proved with common facts, as the district court demonstrated: it found that 20 percent of preschoolers referred for a disability evaluation received an eligibility determination after the statutory deadline, if it all. And here, too, a single injunction can remedy the harm: the court required the District to meet its statutory deadline 95 percent of the time and to improve its performance by 10 percent in the first year and 5 percent each year thereafter.

The same goes for subclass four, toddlers denied smooth and effective transitions to preschool. Those children claimed that the District's policies and practices prevented them from entering preschool by age three without interruption in their special education services. This is a common allegation, provable by evidence showing that the District failed to provide smooth transitions to 30 percent of toddlers, and remediable by a single injunction requiring annual improvement.

We have no need to belabor the point. These three subclasses are far more precise than the class this court vacated in 2013, whose members shared only the contention that they had been denied FAPEs at some point in their experiences with the District's special education programs.

Relying on Wal-Mart, 564 U.S. at 352, 131 S.Ct. 2541, in which the Supreme Court observed that female employees had failed to show common reasons for their managers' decisions about promotions and pay, the District argues that even if its policies run afoul of IDEA, "there are many different reasons" it might have denied a particular child a FAPE. Appellants' Br. 39. For example, the District explains that it may have deprived some children of special education because of "insufficient outreach" and others due to "insufficient staff" or "documentation errors." Id. Accordingly, the District claims, even the three subclasses lack "common contention[s]" whose "truth or falsity" can be resolved "in one stroke." Wal-Mart, 564 U.S. at 350, 131 S.Ct. 2541.

There is, however, a significant distinction between Wal-Mart and this case. As the Court pointed out in Wal-Mart, "[i]n resolving an individual's Title VII claim, the crux of the inquiry is `the reason for a particular employment decision.'" Id. at 352, 131 S.Ct. 2541 (quoting Cooper v. Federal Reserve Bank of Richmond, 467 U.S. 867, 876, 104 S.Ct. 2794, 81 L.Ed.2d 718 (1984) (emphasis added)). The fact that Wal-Mart supervisors might have had different reasons for "literally millions of employment decisions" was therefore fatal to the commonality of the plaintiffs' Title VII claims. Id. By contrast, IDEA requires the District to find and serve all children with disabilities as a condition of its funding. See, e.g., 20 U.S.C. § 1412(a)(3)(A). Unlike Title VII liability, IDEA liability does not depend on the reason for a defendant's failure and plaintiffs need not show why their rights were denied to establish that they were. They need only show that the District in fact failed to identify them, failed to provide them with timely eligibility determinations, or failed to ensure a smooth transition to preschool. Wal-Mart's analysis of commonality in the Title VII context thus has limited relevance here.

Citing a Seventh Circuit decision, Jamie S. v. Milwaukee Public Schools, 668 F.3d 481 (7th Cir. 2012), the District argues that the parents in this case alleged only a "superficial violation of the same provision of law." Appellants' Reply Br. 16. But that case concerned a much broader class, more like the original class this court rejected than the subclasses now at issue. As the Seventh Circuit noted, the class there not only sought "individualized relief," but also "combined," among other groups, "all disabled students ... who were not identified as potentially eligible for services, not timely referred for evaluation after identification, [and] not timely evaluated after referral." Id. at 495, 499. The subclasses in this case suffer from no such flaw and seek class-wide injunctive relief.

The District next argues that the class violates Rule 23(a)(3)'s typicality requirement because the parents' claims "vary from child-to-child." Appellants' Br. 42. This argument should sound familiar: it is the District's commonality challenge in a new guise. Indeed, the District expressly claims that the court "found typicality based on the same error it made in finding commonality." Id. 41. As we have already explained, however, the district court made no such error. Specifically analyzing typicality, it found a "sufficient nexus" between the claims of the named plaintiffs and the claims of the members of their respective subclasses. D.L. v. District of Columbia, 302 F.R.D. 1, 14 (D.D.C. 2013). We see no abuse of discretion.

Rule 23(a)'s final requirement — adequacy — provides that the named plaintiff must "fairly and adequately protect the interests of the class." Amchem Products, Inc. v. Windsor, 521 U.S. 591, 625, 117 S.Ct. 2231, 138 L.Ed.2d 689 (1997) (quoting Fed. R. Civ. P. 23(a)(4)). This rule "serves to uncover conflicts of interest between named parties and the class they seek to represent." Id. In this case, the district court determined that notwithstanding the mootness of their individual claims, the six named plaintiffs "displayed a strong commitment to resolving this case" and "respond[ed] to all developments in a timely and professional fashion." D.L., 302 F.R.D. at 14-16.

Challenging this determination, the District argues that the court "disregard[ed] the presumption" that "[w]hen a plaintiff's claim is moot, it makes her representation presumptively inadequate." Appellants' Br. 42. The court did no such thing. It acknowledged the adequacy concerns raised by the named plaintiffs' age progression and explained why they nonetheless remained capable representatives. The Supreme Court, moreover, has made clear that mootness and adequacy are "separate issue[s]" and that plaintiffs with moot claims may adequately represent a class. Geraghty, 445 U.S. at 407, 100 S.Ct. 1202. With the benefit of firsthand exposure to the parents and their lawyers during the course of a then-eight-year-old case, the district court found that the parents will "fairly and adequately protect the interests of the class." D.L., 302 F.R.D. at 14 (quoting Fed. R. Civ. P. 23(a)(4)). The District has given us no basis for questioning that decision, especially given that the district court pondered and rejected the exact arguments the District now makes.

Nor, contrary to the District's argument, did the court err in certifying subclasses under Rule 23(b)(2). To certify a class under this provision, a single injunction must be able to "provide relief to each member of the class." Wal-Mart, 564 U.S. at 360, 131 S.Ct. 2541. The district court's comprehensive order does just that. Rule 23(b)(2) exists so that parties and courts, especially in civil rights cases like this, can avoid piecemeal litigation when common claims arise from systemic harms that demand injunctive relief. See Geraghty, 445 U.S. at 403, 100 S.Ct. 1202 (discussing the purpose of Rule 23); In re District of Columbia, 792 F.3d 96, 102 (D.C. Cir. 2015) ("Rule 23(b)(2) was intended for civil rights cases."). The Rule 23(b)(2) class action, in other words, was designed for exactly this sort of suit.

IV.

This brings us to the District's challenges to the injunction. IDEA authorizes courts to grant "such relief as [they] determine[] is appropriate," 20 U.S.C. § 1415(i)(2)(C)(iii), which "entail[s] broad discretion and implicate[s] equitable considerations," Reid ex rel. Reid v. District of Columbia, 401 F.3d 516, 522 (D.C. Cir. 2005) (citation and internal quotation marks omitted). Moreover, "it goes without saying that federal courts must vigilantly enforce federal law and must not hesitate in awarding necessary relief." Horne v. Flores, 557 U.S. 433, 450, 129 S.Ct. 2579, 174 L.Ed.2d 406 (2009).

A.

The District first argues that the injunction rests on two mistakes of law — one concerning subclass three and the other subclass four. "We review the district court's conclusions of law de novo." United States v. Philip Morris, 566 F.3d 1095, 1110 (D.C. Cir. 2009).

Recall that subclass three covers three-to-five-year-olds who did not receive an eligibility determination within 120 days of being referred for an evaluation. To remedy this injury, the court required the District to issue timely determinations in 95 percent of cases and to improve its performance annually until it meets that benchmark. The District argues that this requirement is "contrary to the plain language of IDEA" because it "start[s] the clock" for eligibility determinations at the date of referral rather than the date a parent or guardian consents to a child's evaluation. Appellants' Br. 47. In support, the District emphasizes that IDEA section 1414 requires an eligibility determination "within 60 days of receiving parental consent for the evaluation," but it ignores the rest of the sentence: "or, if the State establishes a timeframe within which the evaluation must be conducted, within such timeframe." 20 U.S.C. § 1414(a)(1)(C)(i)(I) (emphasis added). Exercising just that option, the District passed a law — section 38-2561.02 of the D.C. Code — which requires an eligibility determination "within 120 days from the date that the student was referred for an evaluation or assessment." D.C. Code § 38-2561.02(a)(1) (emphasis added). District law thus starts the clock just when the court did, at referral rather than parental consent. True, the District has amended section 38-2561.02 to use parental consent as the trigger for the eligibility determination timeline. See id. § 38-2561.02(a)(2)(A). But that amendment, by its terms, will not become effective until "July 1, 2017, or upon funding, whichever occurs later." Id.

To be clear, the District still needs parental consent to evaluate a child, see D.C. Code Mun. Reg. tit. 5, § 3005.2 (requiring reasonable efforts to obtain parental consent for disability evaluations), and nothing in the injunction eliminates that requirement. Indeed, the district court emphasized that "the District should not be blamed for an untimely determination if the parent does not reasonably participate in the ... process," and noted that the District could adopt a parental delay policy exempting cases in which parents cannot be reached or decline to consent. D.L., 194 F.Supp.3d at 71 (citing 34 C.F.R. § 300.301(d)(1)). In this way, the injunction excuses the District from compliance where it is unable to meet its deadlines through no fault of its own.

The District brings a separate challenge to the remedy for subclass four — children denied "smooth and effective" transitions from Part C early intervention programs to Part B preschool programs. In order to qualify as "smooth and effective," a transition must begin "not fewer than 90 days before the [toddler's] third birthday," involve Part B and C personnel, and deliver "seamless" services. See 20 U.S.C. § 1412(a)(9); 34 C.F.R. § 303.209(a)(3)(ii), (b)-(f). For this subclass, the court fashioned a remedy familiar from subclass three: a 95 percent compliance target, with incremental improvement required each year. To avoid confusion, the court made clear that the District could report a smooth transition so long as children receive all "special education services" by their third birthdays (or slightly later, for those with weekend, holiday, and summer birthdays) and all "related services" by fourteen days thereafter (or again, later in certain cases). D.L., 194 F.Supp.3d at 101. "Related services" are the "support services required to assist a child" in benefitting from special education. Endrew F., 137 S.Ct. at 994 (citing 20 U.S.C. § 1401(26), (29)) (internal quotation marks omitted). In other words, the court ordered the District to provide core services by age three and support services two weeks later.

The District argues that the court had no authority to order it to provide special education services by a child's third birthday — or for that matter, by any date certain. We disagree. IDEA expressly requires states to ensure that an IEP "has been developed and is being implemented" by "the third birthday" of all toddlers transitioning from early intervention to preschool. 20 U.S.C. § 1412(a)(9). Ignoring this statutory text, the District points to a Department of Education regulation requiring services to be delivered "as soon as possible" after development of a child's IEP. 34 C.F.R. § 300.323(c). The statute, of course, is our guide, and it requires IEPs to be implemented by the child's third birthday. Moreover, the regulation the District invokes, which concerns the provision of special education services to students aged three to twenty-one, has nothing to do with the transition to preschool for children already identified as disabled. And if all this were not enough, the regulation governing the "smooth transition" condition requires — in language identical to the statute — that an IEP "has been developed and is being implemented" by age three, id. § 300.124(b), and a separate regulation obliges states to make a FAPE available "no later than the child's third birthday," id. § 300.101(b)(1)(i).

B.

Next the District next makes a series of evidentiary arguments: that the court chose unduly harsh compliance targets, ignored improvements in the Child Find program, and relied on inaccurate statistics. Our review of such objections is "deferential — clear error as to any factual findings and abuse of discretion as to the remedy." Reid, 401 F.3d at 522.

The first of these challenges concerns subclass one — three-to-five-year-olds with disabilities whom the District failed to find. The district court determined that at least 8.5 percent of preschoolers should be enrolled in special education and related services, but that just over 6 percent were. The court based the 8.5 percent benchmark on expert testimony "related to risk factors in the District, comparisons to other jurisdictions, and incidence of developmental delays nationwide." D.L., 194 F.Supp.3d at 49. Specifically, the court found that although "nationally, about six percent of three-to-five-year-olds are identified with developmental delays," the number is likely higher in the District because of its unique risk factors, including unusually high rates of poverty, child homelessness, single-parent and non-English-speaking households, incidence of low birth weight and HIV/AIDS infection, and participation in supplemental assistance programs. Id. at 50. By way of comparison, the court noted that "other urban jurisdictions" with similar risk profiles such as Atlanta and Detroit identified "between 10 and 12 percent" of children as eligible for special education. Id. at 49. "All of this [evidence]," the court reasoned, "supports the conclusion that the District must show that it is serving 8.5 percent of its population," i.e., the figure at the low end of its expected identification rates. Id. at 51. The court ordered the District to improve its performance half a percent each year until it reaches 8.5 percent. Id.

The District believes that this remedy suffers from three flaws: (1) the 8.5 percent benchmark ignores "protective factors" such as the existence of non-profits, which "buffer children against the negative effects" of the risk factors in the District and therefore reduce the number of children likely to need special education services; (2) enrollment data, which the injunction uses to measure the District's success at finding children, do not approximate Child Find compliance because not all children identified as potentially needing IDEA services ultimately enroll in special education programs; and (3) the injunction improperly defines "enrollment" as receipt of all — rather than only some — services promised in a child's IEP. Appellants' Br. 45.

The district court considered and rejected each of these arguments. It took seriously the existence of protective factors that might drive down special education rates, acknowledging that "Washington D.C.'s network of non-profits likely does indeed help to alleviate some of the negative developmental effects of risk factors like high homelessness and poverty rates." D.L., 194 F.Supp.3d at 52. Even so, the court explained, the impact of non-profits and other city services was "baked into" the numbers the parents' expert had used because, "to the extent that the non-profits do decrease rates of poverty and homelessness, etc.," that decrease was already reflected in the District's rates of homelessness, poverty, and other risk factors. Id. As a result, the court reasoned, the data used to design the remedy "incorporate[d] the positive effects" of D.C.'s network of non-profits. Id. at 53.

As to enrollment rates, the court relied on the fact that the District itself treats enrollment as a proxy for identification when tracking and reporting its IDEA compliance. Adopting the District's own methods hardly amounts to an abuse of discretion; quite to the contrary, it makes perfect sense. Nor, in our view, was it unreasonable for the court to define "enrollment" as provision of all services outlined in a child's IEP. After all, IDEA obliges states not only to find children with disabilities, but also to give them services. See 20 U.S.C. § 1412(a)(1)(A) (requiring a FAPE to be made "available to all children with disabilities"); (a)(2) (setting a goal of "full educational opportunity"); (a)(3) (requiring states to develop and implement effective Child Find policies). As the district court put it, "the entire point of the Child Find requirement is to provide services to children with disabilities," a duty the District is violating by offering children only some of the services to which they are entitled. D.L., 194 F.Supp.3d at 91.

The District also challenges the injunction's "programmatic" requirements, which aim to improve data collection and outreach efforts — for instance, by ordering school officials to "publish printed materials targeted to parents and guardians" about available services and to "maintain and regularly update a list of primary referral sources." Id. at 101. The District insists that these requirements are tailored to harms they addressed in response to the district court's 2011 injunction. Perhaps so, but following a three-day trial with new evidence, the district court found a slew of continuing deficiencies in the District's Child Find program, including "material inconsistencies in the District's documents and practice," evidence that the District had yet to amend policies the court ordered it to change five years earlier, and a two-year decline in the District's identification rates after the first injunction was vacated in 2013. Id. at 99. Given this evidence, we see no abuse of discretion in the district court's decision to again impose programmatic remedies.

Next, the District mounts several challenges to the court's reliance on statistical evidence. Several of these arguments are, in essence, objections to the liability finding. For example, the District contends that "the generalized and procedural nature of the findings does not support the conclusion that children were denied a FAPE." Appellants' Br. 57. Other critiques focus on the statistics the court used, which the District asserts are inadequate to support "systemic relief." Id. 54-55. In both formulations, the District's challenge fails.

Not only does the District cite nothing for the proposition that courts may not rely on statistical evidence, but it makes perfect sense to use such evidence where, as here, the violations amount to a systemic failure to find children. How else could the court have demonstrated a failure to identify children with disabilities except with numerical evidence that the District is in fact failing to find and serve specified numbers of such children? And how else could the court have remedied that violation except by setting numerical goals to bring the District into compliance with its IDEA obligations?

Moreover, the District's violation was clearly substantive. Although it is true that "this court has at times required parents to demonstrate that the student's education was affected by any procedural violations the school district might have committed," we have done so "only where the violation was not obviously substantive." Leggett v. District of Columbia, 793 F.3d 59, 67 (D.C. Cir. 2015) (quoting Lesesne ex rel. B.F. v. District of Columbia, 447 F.3d 828, 834 (D.C. Cir. 2006)) (alterations and internal quotation marks omitted). The District asserts that its failure to locate disabled children is a "procedural" rather than substantive harm. Appellants' Br. 57. But twice in recent months the Supreme Court has confirmed that access to a FAPE is a "substantive right." Endrew F., 137 S.Ct. at 993; see also Fry v. Napoleon Community Schools, ___ U.S. ___, 137 S.Ct. 743, 749, 197 L.Ed.2d 46 (2017) ("An eligible child ... acquires a substantive right to [a FAPE] once the state accepts the IDEA's financial assistance." (citation and internal quotation marks omitted)). Disabled children are quite obviously denied a FAPE when the District fails to find them at all.

C.

Finally, we reach the District's two deepest objections to the injunction.

First, the District contends that the court "should have erred on the side of leaving control of the school system to state and local authorities." Appellants' Br. 46. This argument completely ignores the court's restrained approach. Rather than "assum[ing] control" of the District, id. 55, the court opted for benchmarks and gradual deadlines, leaving the District with flexibility in how to achieve compliance and time to do so. As the parents explain, the injunction "require[s] [the District] to do nothing more than what is required under IDEA," and the "programmatic requirements are limited to the basic elements of an adequate Child Find program." Appellees' Br. 56. The injunction balances the need for relief with deference to school administrators, precisely what the court is supposed to do. See Freeman v. Pitts, 503 U.S. 467, 489, 112 S.Ct. 1430, 118 L.Ed.2d 108 (1992) ("[T]he court's end purpose must be to remedy the violation and, in addition, to restore state and local authorities to the control of a school system....").

Second, the District argues that the court "strayed impermissibly from the focus on individual relief that is [at] the heart" of IDEA. Appellants' Br. 56 (citing 20 U.S.C. § 1415). As the District sees it, the "proper role" of IDEA's judicial enforcement provision "is individualized rather than systemic relief." Id. 55. In other words, the District believes that IDEA precludes comprehensive injunctions. This is the last iteration of an argument the District has pressed throughout this litigation: IDEA claims ought to be handled one-by-one, not as class actions cured through structural remedies.

It is true that courts may remedy certain IDEA disputes, such as a parent's claim that a child's IEP is defective, only through "individualized" relief. See Jamie S., 668 F.3d at 495. But to argue, as does the District, that this limitation also applies to violations of the Child Find requirement ignores that, unlike a parent worried about her child's IEP, the parents in this case challenge systemic defects in the District's identification and eligibility determination policies, which harm all unidentified preschoolers and can only be remedied by a comprehensive injunction designed to bring the District into compliance with IDEA. See supra Part III.

Even more important, the District's argument would eviscerate the very purpose of IDEA. When Congress enacted the legislation that became IDEA, it was responding to the "pervasive and tragic" failure to serve all children with disabilities, Endrew F., 137 S.Ct. at 999, which is why it imposed on states accepting IDEA funding an obligation to "identif[y], locate[], and evaluate[]" all preschoolers with disabilities, 20 U.S.C. § 1412(a)(3)(A). Yet the District, which has enthusiastically accepted millions of dollars in IDEA funding, now proposes to shift that burden back to the parents. In the District's view, it would be up to each and every parent, many of whom are poor, homeless, and perhaps disabled themselves, to somehow determine whether their children are eligible for special education services and then to retain counsel to sue the District to obtain the services to which they are entitled. Given the purpose of IDEA, we cannot imagine a more preposterous argument. And given the district court's finding that the District has failed, year after year, to comply with IDEA's Child Find requirement, we have no doubt that the statute's remedial provision — authorizing courts to "grant such relief as [they] determine[] is appropriate," 20 U.S.C. § 1415(i)(2)(C)(iii), and implicating "broad discretion" and "equitable considerations," Reid, 401 F.3d at 522 — vests the court with all the authority it needs to remedy those violations through injunctive relief. For decades, courts across the country have done just that, ordering or approving structural relief when IDEA violations required it. See, e.g., Vaughn G. v. Amprey, 117 F.3d 1415, at *1 (4th Cir. 1997) (recounting the "decade long struggle" between students and the Baltimore City Public School (BCPS) system, which led to a series of consent decrees restructuring BCPS's special education programs); D.D. v. New York City Board of Education, No. 03-cv-2489, ECF No. 250 (E.D.N.Y. Apr. 25, 2007) (approving a decree requiring the New York State Department of Education to amend its policies, coordinate with other agencies, develop and implement training programs, collect data, and propose new legislation to New York State Assembly); Blackman v. District of Columbia, No. 97-cv-1629, 2006 WL 2456413 (D.D.C. Aug. 24, 2006) (approving an expansive consent decree to remedy systemic IDEA violations in the District of Columbia); James O. v. Marston, No. 86-cv-0006, ECF Nos. 191-200 (D.N.H. Aug. 23, 1991) (requiring the New Hampshire Department of Education to overhaul its policies and procedures concerning children with disabilities in detention centers and other state facilities).

V.

Having considered each of the District's challenges, we are convinced that the district court made no mistake. So long as the District of Columbia accepts federal funding, it is bound to its pledge to find, evaluate, and serve all children with disabilities. The district court neither erred nor abused its discretion in holding the District to its word. We affirm in all respects.

So Ordered.